“The design space of plane elastic curves” by Hafner and Bickel

  • ©Christian Hafner and Bernd Bickel




    The design space of plane elastic curves



    Elastic bending of initially flat slender elements allows the realization and economic fabrication of intriguing curved shapes. In this work, we derive an intuitive but rigorous geometric characterization of the design space of plane elastic rods with variable stiffness. It enables designers to determine which shapes are physically viable with active bending by visual inspection alone. Building on these insights, we propose a method for efficiently designing the geometry of a flat elastic rod that realizes a target equilibrium curve, which only requires solving a linear program.We implement this method in an interactive computational design tool that gives feedback about the feasibility of a design, and computes the geometry of the structural elements necessary to realize it within an instant. The tool also offers an iterative optimization routine that improves the fabricability of a model while modifying it as little as possible. In addition, we use our geometric characterization to derive an algorithm for analyzing and recovering the stability of elastic curves that would otherwise snap out of their unstable equilibrium shapes by buckling. We show the efficacy of our approach by designing and manufacturing several physical models that are assembled from flat elements.


    1. Marco Attene, Marco Livesu, Sylvain Lefebvre, Thomas Funkhouser, Szymon Rusinkiewicz, Stefano Ellero, Jonás Martínez, and Amit Haim Bermano. 2018. Design, Representations, and Processing for Additive Manufacturing. Synthesis Lectures on Visual Computing 10, 2 (2018), 1–146. Google ScholarCross Ref
    2. B. Audoly and Y. Pomeau. 2010. Elasticity and Geometry: From Hair Curls to the Nonlinear Response of Shells. Oxford University Press. https://books.google.at/books?id=FMQRDAAAQBAJGoogle Scholar
    3. Milan Batista. 2015. On stability of elastic rod planar equilibrium configurations. International Journal of Solids and Structures 72 (2015), 144 — 152. Google ScholarCross Ref
    4. Miklós Bergou, Max Wardetzky, Stephen Robinson, Basile Audoly, and Eitan Grinspun. 2008. Discrete elastic rods. In ACM Transactions on Graphics. ACM Press. Google ScholarDigital Library
    5. Amit H. Bermano, Thomas Funkhouser, and Szymon Rusinkiewicz. 2017. State of the Art in Methods and Representations for Fabrication-Aware Design. Computer Graphics Forum 36, 2 (2017), 509–535. Google ScholarDigital Library
    6. Florence Bertails, Basile Audoly, Marie-Paule Cani, Bernard Querleux, Frédéric Leroy, and Jean-Luc Lévêque. 2006. Super-helices for predicting the dynamics of natural hair. ACM Transactions on Graphics 25, 3 (2006), 1180–1187. Google ScholarDigital Library
    7. Florence Bertails-Descoubes, Alexandre Jourdan, Victor Romero, and Arnaud Lazarus. 2018. Inverse design of an isotropic suspended Kirchhoff rod: theoretical and numerical results on the uniqueness of the natural shape. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 474, 2212 (2018), 20170837. Google ScholarCross Ref
    8. Gaurav Bharaj, David I. W. Levin, James Tompkin, Yun Fei, Hanspeter Pfister, Wojciech Matusik, and Changxi Zheng. 2015. Computational Design of Metallophone Contact Sounds. ACM Transactions on Graphics 34, 6, Article 223 (2015), 13 pages. Google ScholarDigital Library
    9. Bernd Bickel, Paolo Cignoni, Luigi Malomo, and Nico Pietroni. 2018. State of the Art on Stylized Fabrication. Computer Graphics Forum 37, 6 (2018), 325–342. Google ScholarCross Ref
    10. Max Born. 1906. Untersuchungen über die Stabilität der elastische Linie in Ebene und Raum, unter verschiedenen Grenzbedingungen. Göttingen, Dieterich’sche Univ.-Buchdruckerei.Google Scholar
    11. S.B. Coşkun. 2010. Analysis of Tilt-Buckling of Euler Columns with Varying Flexural Stiffness Using Homotopy Perturbation Method. Mathematical Modelling and Analysis 15, 3 (2010), 275–286. Google ScholarCross Ref
    12. Pierre Cuvilliers, Paul Mayencourt, and Caitlin Mueller. 2018. The Arc Lamp workshop at AAG 2018: active bending and digital fabrication. http://digitalstructures.mit.edu/page/blog#the-arc-lamp-workshop-at-aag-2018-active-bending-and-digital-fabrication Accessed: 2021-01-27.Google Scholar
    13. Alexandre Derouet-Jourdan, Florence Bertails-Descoubes, and Joëlle Thollot. 2010. Stable Inverse Dynamic Curves. ACM Transactions on Graphics 29 (2010). Google ScholarDigital Library
    14. Levi H Dudte, Etienne Vouga, Tomohiro Tachi, and Lakshminarayanan Mahadevan. 2016. Programming curvature using origami tessellations. Nature materials 15, 5 (2016), 583–588.Google Scholar
    15. Moritz Fleischmann and Achim Menges. 2011. ICD/ITKE Research Pavilion: A Case Study of Multi-disciplinary Collaborative Computational Design. In Computational Design Modelling. Springer Berlin Heidelberg, 239–248. Google ScholarCross Ref
    16. Akash Garg, Andrew O. Sageman-Furnas, Bailin Deng, Yonghao Yue, Eitan Grinspun, Mark Pauly, and Max Wardetzky. 2014. Wire mesh design. ACM Transactions on Graphics 33, 4 (2014), 1–12. Google ScholarDigital Library
    17. Konstantinos Gavriil, Ruslan Guseinov, Jesús Pérez, Davide Pellis, Paul Henderson, Florian Rist, Helmut Pottmann, and Bernd Bickel. 2020. Computational Design of Cold Bent Glass Façades. ACM Transactions on Graphics 39, 6, Article 208 (2020), 16 pages.Google ScholarDigital Library
    18. I. M. Gelfand and S. V. Fomin. 1963. Calculus of Variations. Prentice Hall.Google Scholar
    19. Ruslan Guseinov, Connor McMahan, Jesús Pérez, Chiara Daraio, and Bernd Bickel. 2020. Programming temporal morphing of self-actuated shells. Nature communications 11, 1 (2020), 1–7.Google Scholar
    20. Ruslan Guseinov, Eder Miguel, and Bernd Bickel. 2017. CurveUps. ACM Transactions on Graphics 36, 4 (2017), 1–12. Google ScholarDigital Library
    21. Liang He, Huaishu Peng, Michelle Lin, Ravikanth Konjeti, François Guimbretière, and Jon E. Froehlich. 2019. Ondulé: Designing and Controlling 3D Printable Springs. In Proceedings of the 32nd Annual ACM Symposium on User Interface Software and Technology. ACM. Google ScholarDigital Library
    22. Alexandra Ion, Michael Rabinovich, Philipp Herholz, and Olga Sorkine-Hornung. 2020. Shape approximation by developable wrapping. ACM Transactions on Graphics 39, 6 (2020), 1–12. Google ScholarDigital Library
    23. David Jourdan, Melina Skouras, Etienne Vouga, and Adrien Bousseau. 2020. Printing-on-Fabric Meta-Material for Self-Shaping Architectural Models. In Advances in Architectural Geometry. http://www-sop.inria.fr/reves/Basilic/2020/JSVB20Google Scholar
    24. Jan Knippers, Florian Scheible, Matthias Oppe, and Hauke Jungjohann. 2012. Bioinspired Kinetic GFRP-façade for the Thematic Pavilion of the EXPO 2012 in Yeosu. In Proceedings of the IASS-APCS-Symposium, IASS (Ed.). Seoul, Korea.Google Scholar
    25. Mina Konaković-Luković, Julian Panetta, Keenan Crane, and Mark Pauly. 2018. Rapid deployment of curved surfaces via programmable auxetics. ACM Transactions on Graphics 37, 4 (2018), 1–13. Google ScholarDigital Library
    26. Joon Kyu Lee and Byoung Koo Lee. 2018. Elastica and buckling loads of nonlinear elastic tapered cantilever columns. Engineering Solid Mechanics 6 (2018), 39–50. Google ScholarCross Ref
    27. Julian Lienhard, Holger Alpermann, Christoph Gengnagel, and Jan Knippers. 2013. Active Bending, a Review on Structures where Bending is Used as a Self-Formation Process. International Journal of Space Structures 28, 3-4 (2013), 187–196. Google ScholarCross Ref
    28. Mingchao Liu, Lucie Domino, and Dominic Vella. 2020. Tapered elasticæ as a route for axisymmetric morphing structures. Soft Matter 16 (2020), 7739–7750. Issue 33. Google ScholarCross Ref
    29. J. H. Maddocks. 1981. Analysis of nonlinear differential equations governing the equilibria of an elastic rod and their stability. Ph.D. Dissertation. University of Oxford.Google Scholar
    30. Luigi Malomo, Jesús Pérez, Emmanuel Iarussi, Nico Pietroni, Eder Miguel, Paolo Cignoni, and Bernd Bickel. 2019. FlexMaps. ACM Transactions on Graphics 37, 6 (2019), 1–14. Google ScholarDigital Library
    31. Robert S. Manning, Kathleen A. Rogers, and John H. Maddocks. 1998. Isoperimetric Conjugate Points with Application to the Stability of DNA Minicircles. Proceedings: Mathematical, Physical and Engineering Sciences 454, 1980 (1998), 3047–3074. http://www.jstor.org/stable/53424Google Scholar
    32. Eder Miguel, Mathias Lepoutre, and Bernd Bickel. 2016. Computational design of stable planar-rod structures. ACM Transactions on Graphics 35, 4 (2016), 1–11. Google ScholarDigital Library
    33. Dinesh K. Pai. 2002. STRANDS: Interactive Simulation of Thin Solids using Cosserat Models. Computer Graphics Forum 21, 3 (2002), 347–352. Google ScholarCross Ref
    34. J. Panetta, M. Konaković-Luković, F. Isvoranu, E. Bouleau, and M. Pauly. 2019. X-Shells: A New Class of Deployable Beam Structures. ACM Transactions on Graphics 38, 4 (2019), 1–15. Google ScholarDigital Library
    35. Jesús Pérez, Miguel A. Otaduy, and Bernhard Thomaszewski. 2017. Computational design and automated fabrication of kirchhoff-plateau surfaces. ACM Transactions on Graphics 36, 4 (2017), 1–12. Google ScholarDigital Library
    36. Jesús Pérez, Bernhard Thomaszewski, Stelian Coros, Bernd Bickel, José A. Canabal, Robert Sumner, and Miguel A. Otaduy. 2015. Design and fabrication of flexible rod meshes. ACM Transactions on Graphics 34, 4 (2015), 1–12. Google ScholarDigital Library
    37. Stefan Pillwein, Kurt Leimer, Michael Birsak, and Przemyslaw Musialski. 2020. On elastic geodesic grids and their planar to spatial deployment. ACM Transactions on Graphics 39, 4 (2020). Google ScholarDigital Library
    38. Yuri Sachkov and Stanislav Levyakov. 2010. Stability of inflectional elasticae centered at vertices or inflection points. Proceedings of the Steklov Institute of Mathematics 271 (2010), 177–192. Google ScholarCross Ref
    39. Oded Stein, Eitan Grinspun, and Keenan Crane. 2018. Developability of triangle meshes. ACM Transactions on Graphics 37, 4 (2018), 1–14. Google ScholarDigital Library
    40. Ioanna Symeonidou. 2015. Analogue and digital form-finding of bending rod structures. In Proceedings of IASS Annual Symposia, Vol. 2015. International Association for Shell and Spatial Structures (IASS), 1–12.Google Scholar
    41. Nico Van der Aa, H.G. Morsche, and R.R.M. Mattheij. 2007. Computation of eigenvalue and eigenvector derivatives for a general complex-valued eigensystem. The Electronic Journal of Linear Algebra 16 (2007). Google ScholarCross Ref
    42. Guanyun Wang, Ye Tao, Ozguc Bertug Capunaman, Humphrey Yang, and Lining Yao. 2019. A-line: 4D Printing Morphing Linear Composite Structures. In Proceedings of the 2019 CHI Conference on Human Factors in Computing Systems. ACM. Google ScholarDigital Library
    43. Katja Wolff, Roi Poranne, Oliver Glauser, and Olga Sorkine-Hornung. 2018. Packable Springs. Computer Graphics Forum 37, 2 (2018), 251–262. Google ScholarCross Ref
    44. Hongyi Xu, Espen Knoop, Stelian Coros, and Moritz Bächer. 2019. Bend-it: Design and Fabrication of Kinetic Wire Characters. ACM Transactions on Graphics 37, 6 (2019), 1–15. Google ScholarDigital Library
    45. Xu Zheng, Zhichao Fan, Haoran Fu, Yuan Liu, Yanyang Zi, Yonggang Huang, and Yihui Zhang. 2019. Optimization-Based Approach for the Inverse Design of Ribbon-Shaped Three-Dimensional Structures Assembled Through Compressive Buckling. Physical Review Applied 11 (2019). Google ScholarCross Ref

ACM Digital Library Publication: